Such a time-continuous wave as music or voice is called a continuous signal. Though these signals have the value according to the progress of time, a signal is also called a continuous signal that its value varies continuously according to distance or something else. And such signal processing is called continuous signal processing that is executed continuously to a continuous signal. Such a signal is also a continuous signal that its value is continuous in space like a picture. Such signal processing is also called continuous signal processing that it is executed continuously in space to a continuous signal on the space. Such a signal is a one-dimensional continuous signal that one variance as time or distance decides its continuous value. Such a signal is a multi-dimensional continuous signal that its value is continuous in space like a picture
On the other hand, a discrete time sequence like a digital watch or a discrete spatial point sequence like a chessboard is called discrete sample points or sample points simply. The value defined at each discrete sample point is called a sample value. A sequence of sample values is called a discrete signal. Also in this case, when the discrete sample points are placed on one axis, it is called a one-dimensional discrete signal. When the sample values are defined in a multi-dimensional space, it is called a multi-dimensional discrete signal. A sample value may be a sample value of some continuous signal or it may be simply a numerical value defined at a sample point. Especially, an operation to obtain sample values of a continuous signal, is called sampling. Also, operations that obtain a sample value or a part of sample values or all the sample values of a discrete signal at discrete sample points, are called sampling. A discrete sample value may be an irrational number such as τ, for example. Generally, it is not always a finite digit binary number. Therefore, a sample value would be truncated to obtain finite digit binary number for convenience of calculation or data compression. That process is called quantization.
Suppose that there are two signals f and h. A signal y is defined as an output signal of a system S when the signal f is input to the system S. A signal z is defined as an output signal of a system S when the signal h is input to the system S. That is, y=S(f) and z=S(h). When the sum of the signals f and h, that is, (f+h), is input to the system S, suppose that the output signal of the system S becomes always (y+z), that is, the sum of two outputs y and z. Then, the system S is called a linear system. That is to say, if the following equation is valid, the system S is called a linear system.S(f+h)=(y+z)=S(f)+S(h)Let a and b be complex constants. In general, if the following equation is valid, the system S is called a linear system.S(af+bh)=(ay+bz)=aS(f)+bS(h)A system that is not linear is called a non-linear system. An output signal of a non-linear system does not always become double when an input signal is made double. An ordinary communication filter, a lens or a camera can be considered as a linear system. However, a communication filter becomes a non-linear system when its output signal is quantized, for example. Strictly saying, depending upon the characteristics of materials, a lens or a camera may become a non-linear system.
A signal can be divided into a set of sinusoidal waves of different frequencies in many cases. The sequence of the different sinusoidal waves to form the signal is called a frequency spectrum or a frequency characteristic of the signal. Such mathematical conversion is called Fourier transform to derive the frequency spectrum of the signal from the signal itself. The transformation to carry out inverse operation is called inverse Fourier transform. The Fourier transform and inverse Fourier transform are defined to a signal having limited energy. The 2nd power of a norm can be considered as a generalized concept of energy. A set of signals with limited squared norm is known well in the mathematical world as Hilbert space. There are many transformation defined in Hilbert space such as Hankel transform used in the analysis of a lens system, a wavelet transform used for a radar and so on. A signal and its spectrum can be defined also in Hilbert space. In that case, both the formulation to derive a spectrum from a signal and an inverse formulation to compose a signal from its spectrum do not necessarily hold simultaneously. For example, in some case, a spectrum is defined first and then a signal would be defined by the inverse transform of the spectrum.
Observing the various systems in the present industrial field and the present medical field, such equipments that perform the following estimation or processing of signals are included in the systems in many cases.
(1) A preprocessing is performed to obtain unknown external signals through a given linear or non-linear system or through given parallel divided linear or non-linear systems.
(2) The preprocessed signal is sampled at discrete sample points to obtain the sample value of the signal. All or some of the sample values are quantized in postprocessing to create discrete data.
(3) This discrete data is processed in certain way to estimate the unknown external signal.
At this time, the system for continuous signals like a camera, an optical device, an analog filter, computer tomography or NMR equipment may be used as an acquisition system for unknown signals stated in (1). There is also a system like a discrete filter that is used in a digital signal processing or in a time-sharing communication system, in order to preprocess a signal after digitizing the signal.
However, the processing is mainly discrete signal processing in many cases according to recent rapid progress of LSI technology. For this reason, these equipments are almost as above-mentioned signal-processing system. That is, discrete data is obtained by sampling those signals whether the preprocessed signal is a continuous signal or a discrete signal. In many cases, the unknown signal is approximately estimated by discrete signal processing based on those discrete data. Some equipments use all the discrete data and some equipments use discrete data partially. The final estimated signal itself, might be a discrete signal. The final estimated signal may be obtained, by converting the approximate discrete signal that is obtained by the signal-estimation into continuous signal, by interpolation.
Number of discrete data and interval of sample points relate directly to the computational complexity for post-processing. The importance of signal processing has increased in recent years in natural image processing, medical image processing, image processing at SAR radar and image processing a robot's eye. The sample points for processing increase instantly and many expensive high-speed LSIs are required when the sample point interval is shortened to estimate the signal accurately in such systems with multi-dimensional signal processing as mentioned above. For this reason, sample values are thinned out in creating data. The input signals between the remained thinned-out samples are approximately restored with interpolation. Or, the input signals are approximately restored with extrapolation to predict the input signals outside of the sampling region. There arises a problem to minimize the approximation error due to interpolation or extrapolation.
Method to approximate the original signal with linear processing of sample values, is explained. Consider such function or vector function in Hilbert space that the weighted norm of the function is less than given positive number. We call these functions spectrum. Simply saying, the spectrum of lower energy than the specific value is treated. This is practically appropriate. Continuous or discrete signal derived from this spectrum by linear inverse transform, such as sum of product, integral, or linear inverse transform that is generalization of those transformations, is an object signal that is processed in approximation. It is assumed that such a signal is given as an external unknown signal and a series of following processing is performed.
(1) External unknown signals are obtained through the preprocessing system.
(2) The sample values of the preprocessed signals are obtained by sampling at the discrete sample points. The discrete data is created with post-processing by quantizing those sample values and so on.
(3) The external unknown signals are estimated by some signal processing from these discrete data.
In the processing of (3), formula that calculates the total or partial sum of each term that is product of each sample value and a function corresponding to coefficient is used as approximation formula. That function multiplied to each sample value is called interpolation function. Considering many kinds of functions to express the error between the resultant approximation and the source signal, we consider the upper limit of those functions with respect to all the source signals as measure of error. There are many kinds of measures of errors such as measure of error of absolute value, measure of error of square area, and so on.
Profile of linear signal preprocessing with linear system is explained. First, an FIR filter is explained. FIG. 5 shows a circuit called FIR filter. As shown in FIG. 5, the input signal f(t) passes the advancing circuit to advance the signal by unit time τ, or the delay circuit to delay the signal by unit time τ. After that, complex coefficient kn (n=−N, −N+1, . . . , N) is multiplied to the signal. More over, the result is totally summed up with an adder circuit to yield the output f0(t).
Here, the limit of the above-mentioned total sum is determined according to the number of the advancing circuit and the delay circuit. In fact, as the advancing circuit is impossible, the output signal is delayed according to the number of the advancing circuits. Therefore, in fact, the signal enters to the leftmost terminal of the circuit shown in FIG. 5. In this reason, when the signal is processed with FIR filter, some delay is inevitable. This delay is of course according to the number of the advancing circuits. The larger is this number, the larger is the delay. Here, ignoring the above-mentioned delay, consider the FIR filter of advancing circuit and delay circuit as shown in FIG. 5.
The unit delay circuit to delay a signal by a unit time τ is considered. The following signal is introduced to this circuit.f(t)=exp(jωt) (j: imaginary unit)Then naturally the signal is delayed by a unit time τ and the output signal becomes as follows.
                              f          ⁡                      (                          t              -              τ                        )                          =                  exp          ⁢                      {                          j              ⁢                                                          ⁢                              ω                ⁡                                  (                                      t                    -                    τ                                    )                                                      }                                                  =                              exp            ⁡                          (                              -                jωτ                            )                                ⁢                      exp            ⁡                          (                              j                ⁢                                                                  ⁢                ω                ⁢                                                                  ⁢                τ                            )                                                              =                              exp            ⁡                          (                              -                jωτ                            )                                ⁢                      f            ⁡                          (              t              )                                          That is, the complex sine wave f(t) is multiplied by exp(−jΩτ). This means that transfer function of the unit delay circuit in the angular frequency domain is nothing less than exp (−jΩτ). Therefore, the transfer function of the circuit to delay a signal by t=nτ is exp(−jnωτ). From this fact, it can be realized directly that the transfer function of FIR filter shown in FIG. 5 is as follows.H(ω)=Σn=−NNkn·exp(−jnωτ)
Therefore, the transfer function of FIR filter turns into complex Fourier series. Complex Fourier series can approximate almost all functions in the frequency band of −π<ωτ<π in sufficient accuracy in a practical range if the limit N of total is increased and coefficients kn (n=−N, −N+1, . . . , N) are chosen well. Therefore, FIR filter can be either a low-pass filter or a band-pass filter in the frequency band of −π<ωτ<π according to the chosen tap number and coefficients of the FIR filter.
As the transfer function of FIR filter is Fourier series, the transfer function of FIR filter is a periodic function of 2π period having the fundamental period in the frequency band of −π<ωτ<π. In the digital signal processing, the frequency band of the fundamental period is mainly used. Therefore, the input signal f(t) is also the band-bound signal of Fourier spectrum bound in −π<ωτ<π or the band-bound signal approximately bound in this band. However, this condition is not necessarily satisfied if the error accompanied with approximation can be accepted. The error may become large depending upon the difference if the condition is quite different.
FIR filter bank is explained. FIR filter bank is a system as shown in FIG. 6 as an array of FIR low-pass filters, band-pass filters or high-pass filters. The analysis filter and synthesis filter are FIR filters, respectively. The analysis filter sometimes works to divide the input signal f(t) into several bands of narrow frequency region called sub-band. The sub-band is sometimes used for data compression for example in the case of calculation saving for digital numerical processing. For example, the output signal of each sub-band is approximately represented in given finite digit binary fractions etc. If the output of one sub-band is small for some time sequence, it is approximately turned to 0. Thus the data is compressed. As FIR filter can realize many kinds of transfer functions, the transfer functions of observing instruments and sensors can be represented using the models of analysis filters as shown in FIG. 6.
Output signal passed through the analysis filter is sampled at each sample point t=nT (n: integer) arranged periodically on the time axis with interval T. Thus the sample value fm(nt) (n: integer, m: pass number) is obtained. These sample values are partial information on f(t). Therefore, in general, the sample values cannot determine f(t) uniquely. Under some condition, the sample values can determine f(t) uniquely. Such condition and the restoring formula from g(t) to f(t) is the sampling theorem. Here, if without notice, it is assumed that the sample values cannot determine f(t).
At the sample points corresponding to these sample values or part of them, the sample value is entered into FIR filter of synthesis filter in turn. The resultant output of the synthesis filter is summed up with the adder circuit at last and then it becomes the final output g(t). The output g(t) is required to be approximated close to the input f(t) as possible. The following fact is self-evident but important. If the input f(t) is known only one, without such complicated approximation as this, f(t) can be presented as g(t) previously. Naturally, the approximation error is as follow.e(t)=f(t)−g(t)=f(t)−f(t)=0Such a trivial case is not treated here. On the contrary, if f(t) can vary much, f(t) cannot be restored with the sample values of partial information of f(t). That is, the function value at the sample point becomes the sample value. But, there are many signals that have values of very big change at the other points than the sample points. Therefore, it is actually impossible to approximate such signals.
Therefore, it is required to set up a suitable set of signal f(t) and to consider the optimal approximation to the set. Even though an approximate formula is very good, if it is good only for a known specific signal, the argument will relapse into the above-mentioned argument. It should be considered to improve approximation performance on the whole to a signal f(t) belonging to a determined signal set.
Then, the target measure and problem of a maximum error are described a little more correctly. Now suppose that a signal set and a certain approximate function for the signal belonging to the signal set are given. The signal set is written as Ξ. The approximation result is written as g(t). The approximation error is written as e(t). Further, suppose that the approximation formula is to approximate the signal f(t) using the sample value fm(nT) of the output fm(t) of the analysis filter Hm(ω) with input signal f(t). Here, m=0, 1, 2, . . . , M−1. n=Nm1, Nm1+1, Nm1+2, . . . , Nm2. Nm1 and Nm2 are given integers defined according to the number m of the analysis filter Hm(ω). Nm1 and Nm2 may not be dependent on the number m of the analysis filter Hm(ω). Then let Nm1=N1 and Nm2=N2. Under that condition, the measure of a maximum error is defined as follows.
The measure of the maximum error: the given signal set is written as Ξ. The approximation result is written as g(t). The approximation error is written as e(t). Moreover, suppose that a positive function β[e (t)] of error e(t) is defined. The function may generally be an operator or a functional. Here, consider the approximation error e(t) of the signal f(t) for all over the signal f(t) belonging to the signal set Ξ. The maximum of β[e(t)] according to these e(t) is written as follows.Emax(t)=sup{β[e(t)]}(the range of sup is f(t)εΞ)It is called the measure of the maximum error. The measure of the maximum error Emax(t) is defined at each time t.
Here, consider the problem to find the synthesis filter to minimize all of the above-mentioned measure of the maximum error together at the same time when the set Ξ of input signal f(t), the interval of the sample points, the transfer function of the analysis filter and the number of taps of the synthesis filter are given. This approximation formula is called the optimum approximation formula hereafter. Moreover, as this approximation formula minimize all of the measure of the maximum error at the same time, naturally, it minimizes the measure of the maximum error Emas(t) without β as follows.Emax(t)=sup{|e(t)|}(the range of sup is f(t)εΞ)When this approximation formula to minimize the measure of the maximum error can be found uniquely, the approximation formula is exactly the approximation formula itself that is found easily by reason of uniqueness of the approximation formula. Is it actually possible?
Existence of undefeatable approximation under some conditions is explained. Here, it is shown first that the approximation formula to satisfy two conditions is the optimal approximation formula mentioned above. Now, suppose that the approximation formula to satisfy the following two conditions exists to certain signal set Ξ.
Condition 1: When the approximation formula is used, the approximation error e(t) to f(t) in the signal set Ξ belongs to the signal set Ξ. That is, the set Ξ contains the set Ξe of approximation error e(t) corresponding to the approximation formula.
Condition 2: Consider the approximation error e(t) as a signal since e(t) belongs to the signal set Ξ. Suppose that the optimal approximation formula is applied to the signal e(t). The sample value em(nT) of the output em(t) of the analysis filter Hm(ω) is 0 altogether when impressing e(t) to the analysis filter. Here, m=0, 1, 2, . . . , M−1. n=Nm1, Nm1+1, Nm1+2, . . . , Nm2. Nm1 and Nm2 are given integers defined according to the number m of the analysis filter Hm(ω).
Now, it is shown that the given approximation formula is the optimal approximation formula mentioned above when it satisfies the two above-mentioned conditions. FIG. 7 shows the candidates of the optimal approximation formula and other approximation formula. The upper half of FIG. 7 corresponds to the optimal approximation formula. The lower half is other approximation for comparing. As shown in FIG. 7, each approximation formula is g(t) and y(t). Corresponding approximation errors are e=e(t) and E=E(t). Since errors e and E are determined by input f(t), they can be written as e(f, t) and E(f, t).
For simplicity, the optimal approximation is called our approximation and other approximation is called their approximation. At this time, the result shown in FIG. 8 is obtained when the approximation error e=e(t)=e(f, t) of our approximation is input as a signal into the analysis filter. As shown in FIG. 8, according to the condition 2, since the sample value is 0 when the signal f=e is input into the analysis filter, in this case, the resultant data through the analysis filter is nothing. Then, as an appropriate assumption, the approximation formula of their approximation at that time is assumed 0(t)=E(f, t)=E(e, t)≡0. Then, as shown in FIG. 8, it holds about their error that E(e, t)=e(f, t)−0(t)=e(f, t)=e. As a result, the inequality shown
That is, the maximum (upper limit) of their approximation error E=E (f, t) when the signal f=f(t) is varied in the original signal set Ξ does not increase naturally if the range to select f is reduced from Ξ to Ξe form the definition of upper limit. Therefore, the second formula in FIG. 9 holds. On the other hand, from FIG. 8,E(e,t)=e(f,t)−0(t)=e(f,t)=e Therefore, the fourth formula holds. By the way, according to the corresponding relation between the signal f and the error signal e, to search for the upper limit of β[e(t)] changing the signal e=e(t)=e(f, t) in the set of error signal e(f, t) is the same as to search for the upper limit of β[e(t)] changing the signal f=f(t) in the signal set Ξ. Accordingly, the last fifth formula can be obtained. Notice the first formula and the fifth formula. Then, it turns out that our approximation has always the measure of approximation error not greater than their approximation. Thus, it is proved.
It is explained with a concrete example. Now, consider an example to sample merely an input signal at each sample point. In this example, assume the input-and-output characteristic graph of a preprocessing circuit is a straight line of the inclination 1 passing the origin.
Our approximation circuit yields 0 at the sample point when the sample value is 0 and yields the approximation signal of the sour ce signal when the sample value is not 0. That is, at the sample point, the circuit yields the sample value of the signal as it is. At the other point than the sample point, it yields the approximation value as mentioned here. Their approximation circuit yields 0 at the sample point when the sample value is 0. It may yield the value different from the sample value when the sample value is not 0. In this case, as our approximation signal coincides with the source signal at the sample point, the sample values of the approximation error e(t) at the sample points are all 0. Thus, the condition 2 is fulfilled.
When their approximation is applied to this approximation error e(t), as the sample values are all 0, their approximation may be be considered as 0. Then, when their approximation is applied to our approximation error, their approximation error is e(t)−0=e(t) as it is input e(t) minus their approximation formula 0. That is, when the input is e(t), our approximation has the same approximation performance with their approximation. However, when the input is f(t), their approximation may have the approximation error greater than e(t). In this case, it turns out that our approximation could yield better approximation value.
Even though their approximation formula could yield an approximation error less than e(t) concerning the specific input f(t), as the signal f(t)=e(t) exists in the source signal according to the condition 1, if it is selected as the source signal, as their approximation has the error approximation e(t) to such input signal, as far as the upper limit error is adopted, also in this case, our approximation and their approximation have the same upper limit error measure Emax(t)=sup{|e(t)|} (the range is f(t) in the set xi). That is, our approximation has the same or better performance than their approximation in the above-mentioned upper limit error measure.
A general example of preprocessing circuit to perform linear transforms. The input signals f(t) are assumed to be f(τ) f(2τ), . . . , f(Nτ). τ is a positive constant. This is vector f. The preprocessing circuit is a linear transform circuit of vector f. For example, y=Af. But, A is a matrix of M times N. The M elements of y are the output of the sampling circuit. Postprocessing circuit function is z=By. But, B is a matrix of N times M. In this time, if the output of A after passing A, B and A again is the same as the simple output of only A, that is, if the sample value A(BAf) of output BAf is the same as the sample value Af of input f, i.e. ABAf=Af, the condition 2 is fulfilled. The reason is this. As the error of that time is e=f−BAf, the sample value of the error e is A(f−BAf)=(A−ABA)f. If ABA=A, then the sample value of error e becomes 0. Thus the condition 2 is fulfilled. In this time, the condition that the error e=(A−ABA)f is contained in the set xi of f is the condition 1. The problem is that such conditions are realized.
From the above consideration, there arises a subject “Find the practical signal set to satisfy above-mentioned two conditions and the optimal approximation formula under such conditions.” The following signal set is considered as such an example of the first. That is, as shown in FIG. 10, assume that the Fourier spectrum of source signal h(t) is H(ω) and the energy of source signal h(t) is less than or equal to given positive number A. This time, it is assumed that the output is f(t) when the source signal h(t) is entered to the filter with given positive transfer function √(W(ω)) and its Fourier spectrum is F(ω). As shown in FIG. 10, F(ω) will fill the inequality shown in FIG. 10. The set of signal f(t) with F(ω) to fill such restrictions is considered as the signal set xi.
The function W(ω) is called a weight function. Its example is shown in FIG. 11. For example, h(t) is the function to show the rushing air flow in a lung. The square root of W(ω) is the function to show the windpipe of throat. The signal f(t) is the voice out of the mouth. Then the signal approximation can be done when the set of signal f(t) is considered as a set of voice. At this time, ωc in FIG. 11 is corresponding to the approximate cutoff frequency of voice, for example, 4 kHz. In this case, if the sample points are fixed at finite numbers, the optimum approximation formula to satisfy the conditions 1 and 2 can be found. The approximation formula uses the synthesis filter Ψm, n(ω) with different function scheme at each sample point. m is path number. n is the number of sample point. It is different with the circuit in FIG. 6 that the filter is different at each sample point. The optimum filter ωm, n(ω) is restricted in the frequency range of −π<ωτ<π. τ is the sampling period of the source signal. It is almost always 1/M of the sample point interval T. If it is not satisfied, though the approximation is not broken, but the error may increase.
In this case, as the impulse response φm, n(t) of the synthesis filter Ψm, n(ω) has an infinite length, actually, it is used by discontinuing at the finite fixed time width. Therefore, this approximation is the approximation in that time width. In this case, as the impulse response φm, n(t) is restricted in that time width, when the discrete approximation is performed, the impulse response φm, n(t) can be realized as the FIR filter with finite taps. At this time, the measure of the approximation Emax(t)=sup{|e(t)|} (the range of sup is defined as that f(t) is in xi) is expressed in integral as follows.Emax(t)=(√A)/(2π){∫−∞∞W(ω)2×|exp(jωt)−Σm=0M−1Σn=N1N2φm,n(t)Hm(ω)exp(jωnT)|2dω}1/2 Here, φm, n(t) is the impulse response of the optimum synthesis filter.
The method to find the optimum φm, n(t) concretely is as follows. First, the absolute value of the complex function included in the integral of the measure of the above-mentioned upper limit error is developed with rewriting as the product of the complex function and its conjugate complex function. It is re-arranged with regard to φm, n(t). As the terms with regard to m, n(t) are not relevant to omega, all of them are getting out of the integral with regard to omega. Therefore, it can be easily differentiate e with regard to φm, n(t). The above-mentioned measure of error is differentiate with regard to all φm, n(t) in this way and the results are set as 0. Then the result is a set of simultaneous linear equations.
The terms except φm, n(t) in that simultaneous linear equations are proportional to the Fourier inverse transform (or its value at the specific time) of W(ω)Hm(ω)) or the product of W(ω)Hm(ω) and the complex conjugate of Hm(ω). Then, the high-speed calculation method of Fourier inverse transform as FFT can be utilized for it. By solving this simultaneous linear equations, the optimum φm, n(t) can be obtained. Though the detail discussion is omitted, at this time, the coefficient matrix of the concerned simultaneous linear equations becomes the constant matrix. Once it is calculated, then it is sufficient only to memorize the values. It should be noted that the re-calculations at every t interval are not necessary. And also, once someone calculates φm, n(t) itself primarily, others should use only the values at all.
And, the point that the equation to derive φm, n(t) becomes as simultaneous linear equations is very advantageous on numerical computation. Actually, the point that the high-speed calculating method such as LU method is available is an advantageous point of the computer software of this approximation. The fact that the impulse response of the band-restricted synthesizing filter with such many advantages minimizes the measure of maximum error simultaneously all together is very important.
The above-mentioned first example is a result in case the finite number of sample points is fixed. Next, as the second example, the approximation in the case of the usual filter bank using FIR filters as shown in FIG. 6 is explained. The signal set in this case should be corresponding to such signal set in the case that the frequency band shown in FIG. 10 and FIG. 11 is expanded to −(T+M)π≦ωTτ≦(T+M)π. At this time, the impulse responses of the synthesizing filter Ψm(ω) with the following feature are derived. φm(t) (m=0, 1, 2, . . . , M−1) is restricted in the above-mentioned expanded band. At the discrete time t=nτ (n is an integer), φm(nτ) (m=0, 1, 2, . . . , M−1) is 0 at the time over the given finite time width. Therefore, φm(t) (m=0, 1, 2, . . . , M−1) is realized as an FIR filter with finite taps when it is used only at the discrete time t=nτ (n is an integer). Moreover, importantly, it can be demonstrated that this approximation formula to use φm(t) (m=0, 1, 2, . . . , M−1) satisfies the condition 2.
As mentioned above, in this case, as long as the argument is limited to so-called discrete approximation that is aimed at the approximation in discrete time t=nτ (n is an integer), it is focused on whether there is any practical signal set that satisfies conditions 1. The sufficiently practical signal set is actually obtained in the form extended to the multi-dimension signal.
The application to digital multiplex communication system is explained. The case of signal approximation corresponding to above-mentioned second example is shown as FIG. 12 with the example of 2-pass filter bank. In FIG. 12, the input signal is sampled with the interval T after passing the analysis filter. As the result of it, so-called aliasing phenomenon arises. As shown in the center of FIG. 12, just after the sampling at the midpoint of the filter bank, the overlap of spectrum arises. As the result, as shown in the lower part of FIG. 12, the simultaneous linear equations hold with the variables of F(ω) and F(ω0−ω). Therefore, if it can be solved, F(ω) and F(ω0−ω) are calculated from the overlapped spectrum obtained by sampling at the midpoint of the filter bank. At this time, in order to solve these simultaneous equations, the determinant of a coefficient matrix needs to serve as non-zero. This is called the condition of independence of the analysis filter. The pass number is assumed as 2, but, it is also the same as in the case of more than or equal to 3.
If the analysis filter is independent, from the sample value just after the sampling at the midpoint of the filter bank, the input signal f(t) can be restored. The analysis filter of such filter bank is connected as shown in FIG. 13. Then, as shown in FIG. 13, assumed that such filter bank has application performance excellent enough, the equation f(t)=g(t) holds. Assumed that the similar signal yields similar output, as shown in FIG. 13, the signals shown with red and green for understanding convenience appear well-separated in the output end. Without the leftmost circuit of FIG. 13, the digital multiplex communication system of good approximation can be realized.
The input signal to such digital multiplex communication system is, in FIG. 13, the sample value of the midpoint of the filter bank, it is naturally certain function or functional of f(t). And also, the output of such digital multiplex communication system is, in FIG. 13, the rightmost sample value, it is naturally the same function or functional of g(t). The discussion is of course on the linear system, the difference between input and output at each channel of such multiplex communication system becomes after all the functional of f(t)−g(t). By the way, in the case of above-mentioned second example, this approximation minimizes the upper limit of any functional of e(t)=f(t)−g(t). Therefore, after all, this approximation minimizes the upper limit of any operator, functional or function of the difference between the input and output at each channel of the multiplex communication system. That is, this approximation is also very advantageous in the design of the multiplex communication system. Besides, note that the time pitch of input and output of such multiplex communication system is T.
The analysis filter is, if the condition of independence is fulfilled, able to set up wide enough in band. Its computing software is simple as FFT or linear computing. Therefore, for example, even if the transmission characteristic of a circuit changes at disaster, the changed circuit characteristic is considered to be an analysis filter, by easy calculation, finding the optimal synthetic filter for moment, and the situation can be coped with flexibly. Especially, the digital multiplex communication system to use this approximation is the circuit shown in the right of FIG. 13. Therefore, the rightmost circuit is, as filter bank, the filter used as the analysis filter. And the center circuit is, as such digital communication system, corresponding to input circuit. This is, in the filter bank, corresponding to the synthesizing filter. That is, the optimum design of such digital multiplex communication system is the problem that the transmission filter (corresponding to analysis filter at the filter bank) at the broadcasting station is optimized with given receiving filters (corresponding to the analysis filter at the filter bank) at many users. In this meaning, it is suitable to the object to enable to cope with the situation flexibly, even if the transmission characteristic of a circuit changes at disaster, the changed circuit characteristic is considered to be an analysis filter, by finding the optimal synthetic filter for moment by easy calculation.
This inventor proposed the approximation method to interpolate a multidimensional signal in nonpatent document 1. By this approximation method, an incoming signal is sampled at discrete sample points after passing a preprocessing filter. From the obtained sample value, an incoming signal is presumed in approximation. The error between the incoming signal and the reproduction approximation signal is the minimum with wide variety of measures of error.
This inventor proposed the optimum discrete approximation of band-limited signals in nonpatent documents 2 and 3. In nonpatent document 4, the optimum interpolation and design of linear phase filter banks was proposed. In nonpatent documents 5, 6, 7, and 9, the optimum approximate restoration of multi-dimensional signals was proposed. In nonpatent documents 8, 10, and 11, interpolatory estimation of multi-dimensional orthogonal expansions was proposed. The optimal linear interpolation was proposed in nonpatent documents 12, 13, and 14.    Nonpatent document 1: Takuro Kida: “Theory of Generalized Interpolation Approximation of Multi-Dimensional Signals”, Journal of Signal processing, Vol. 6, No. 1, pp. 3-8, January 2002, No. 2, pp. 71-77, March 2002, No. 3, pp. 137-141, May 2002.    Nonpatent document 2: Y. Kida and T. Kida, “The Optimum Discrete Approximation of Band-Limited Signals without Necessity of Combining the Set of the Corresponding Approximation Errors,” IEICE Trans. Fundamentals, Vol. E85, No. 3, pp. 610-639, March 2002.    Nonpatent document 3: Y. Kida and T. Kida, “The Optimum Discrete Approximation of Band-Limited Signals with an Application to Signal Processing on Internet,” IEICE Trans. Fundamentals, Vol. E82-A, No. 8, pp. 1592-1607, August 1999.    Nonpatent document 4: T. Kida and Y. Kida, “Consideration on the Optimum Interpolation and Design of Linear Phase Filterbanks with High Attenuation in Stop Bands,” IEICE Trans. Fundamentals, Vol. E81-A, No. 2, pp. 275-287, February 1998.    Nonpatent document 5: T. Kida and Y. Zhou, “The Optimum Approximate Restoration of Multi-Dimensional Signals Using the Prescribed Analysis or Synthesis Filter Bank,” IEICE Trans. Fundamentals, Vol. E79-A, No. 6, pp. 845-863, June 1996.    Nonpatent document 6: T. Kida, “The Optimum Approximation of Multi-Dimensional Signals Based on the Quantized Sample Values of Transformed Signals,” IEICE Trans. Fundamentals, Vol. E78-A, No. 2, pp. 208-234, February 1995.    Nonpatent document 7: T. Kida, “On Restoration and Approximation of Multi-Dimensional Signals Using Sample Values of Transformed Signals,” IEICE Trans. Fundamentals, Vol. E77-A, No. 7, pp. 1095-1116, July 1994.    Nonpatent document 8: T. Kida, S. Sa-Nguankotchakorn, K. Jenkins, “Interpolatory Estimation of Multi-Dimensional Orthogonal Expansions with Stochastic Coefficients,” IEICE Trans. Fundamentals, Vol. E77-A, No. 5, pp. 900-916, May 1994.    Nonpatent document 9: T. Kida, “The Optimum Approximation of Multi-Dimensional Signals Using Parallel Wavelet Filter Banks,” IEICE Trans. Fundamentals, Vol. E76-A, No. 10, pp. 1830-1848, October 1993.    Nonpatent document 10:T. Kida, S. Sa-Nguankotchakorn, “Generalized Optimum Interpolatory Estimation of Multi-Dimensional Orthogonal Expansions with Stochastic Coefficients,” IEICE Trans. Fundamentals, Vol. E75-A, No. 12, pp. 1793-1804, December 1992.    Nonpatent document 11:T. Kida, H. Mochizuki, “Generalized Interpolatory Approximation of Multi-Dimensional Signals Having the Minimum Measure of Errors,” IEICE Trans. Fundamentals, Vol. E75-A, No. 7, pp. 794-805, July 1992.    Nonpatent document 12:Takuro KIDA and Hiroshi MOCHIZUKI: “A Systematic Consideration on the Superiority of Generalized Linear Interpolation Approximation,” IEICE Trans., Vol. J75-A, no. 10, pp. 1556-1568, October 1992.    Nonpatent document 13: Takuro KIDA and Somsak SA-NGUANKOTCHAKORN: “Study of Generating Function of Sampling Theorem and Optimum Interpolation Functions Minimizing Some Kinds of Measures,” IEICE Trans., Vol. J74-A, no. 8, pp. 1332-1345, August 1991.    Nonpatent document 14:Takuro KIDA, Leopoldo Hideki YOSHIOKA, Sadayoshi TAKAHASHI and Hajime KANEDA: “Theory on Extended Form of Interpolatory Approximation of Multi-dimensional Waves,” Elec. and Comm. Japan, Part3, 75, no. 4, pp. 26-34, 1992 and IEICE Trans., vol. 74-A, no. 6, pp. 829-839, June 1991.    Nonpatent document 15:Yuichi Kida and Takuro Kida: “Theory of the optimum approximation of vector-signals with some applications,” The 2004 IEEE International Midwest Symposium on Circuits and Systems (MWSCAS 2004). July 2004.